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dderbes
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Ibix said:
This is a tricky thing that most relativists get but that to a first approximation, no popular physics writers understand. (Dr. Hossenfelder is a relativist and understands this very well.)
Let's say we have two clocks at a particular moment at rest with respect to each other, but one is accelerating. Thus far there is NO experimental evidence that the clocks' rates are different. For example, one can consider muons (whose lifetime is very short if they're at rest) moving in straight lines at high velocities, and compare them with muons moving in circular storage rings at Fermilab and elsewhere at the same velocity, and they have the same observed lifetime, (given by the usual time dilation formula involving the square root of ##1 - (v/c)^2##) even though the ones moving in circles are experiencing gigantic accelerations and the others are not. So acceleration all by itself does not affect clock rates. (The hypothesis that acceleration by itself does not influence the clock rate is called "the clock hypothesis". There is no theoretical basis for it thus far that I am aware of.) So far, so good.
However, it is known that clocks on an upper level of an apartment building run at a tiny bit faster than those on the ground level; the higher you go, the faster the clock rate. This is not due to the gravity, exactly (the gravitational field is nearly the same), but to the gravitational potential, which is in this case the product of ##g## and the height, ##y##, divided by ##c^2##. (For folks who remember some physics, gravitational potential means "gravitational potential energy per kilogram".) Google "gravitational red shift" for more about this. The equation comparing clock rates is this:
##\Delta t_{\text{higher}} = \Delta t_{\text{lower}}(1 + (gy/c^2))##
If this effect were not taken account of by GPS satellites, your phone apps telling you where you were would be very badly incorrect!
Finally, there is a thing called "the equivalence principle": if an elevator is small enough, you cannot distinguish between the effects of being accelerated up at ##g## meters per second squared or being stationary (or moving with constant velocity) in the earth's gravitational field of ##g## meters per second squared. What this means is that during acceleration, a person with an accelerated clock would measure the rate of an clock at a distance of ##y## meters away as running <i>faster</i> than hers, according to the formula above. That is, the factor of ##(1 - (v/c)^{2})## has to be replaced by ##((1 + gy/c^{2})^2 - (v/c)^2)## in the square root, where $y$ is the distance between the accelerated clock and the unaccelerated clock.
This is the cheap solution to the so-called "twin paradox". Say a traveling twin sets out from earth, goes to a distant star, turns around and comes back. There are three periods of acceleration: near the earth going away, turning around, and near the earth to slow down. During periods of steady motion, both the traveler and the stay at home see each other's clock running slow with respect to each other's (they have to; this is a theory of relativity). During the acceleration near the earth, both twins agree that the stay at home twin's clock runs faster than the traveler's, but not appreciably so, because the distance ##y## is not very big. However, during the turnaround, the traveler thinks the earth based twin's clock runs much faster, due to the large value of ##y##. Detailed calculation shows that this rate is precisely what is needed (with the slight increase near the earth) to reconcile the two twins' clocks.
All of this was worked out by the Danish physicist Christian M\o ller in 1943 using general relativity, though he didn't realize (apparently) that the result was exact. A French physicist, Henri Arzèlies, went through it and showed indeed that the resolution was exact. In fact, general relativity is not needed at all: the equivalence principle and gravitational time dilation are all that are required.
Reference: https://www.physicsforums.com/threa...time-dilation-is-due-to-acceleration.1051866/
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